Heavy-light mesons, like the \(D\)  and \(B\) , are interesting QCD bound states:
they play an important role in the threshold dynamics of many of the so-called
X,Y,Z exotic hadrons [1]} and also serve as a laboratory for
studying chiral properties of the light \(u\)  and \(d\)  quarks in a medium at finite
temperature \(T\)  and baryon density \(\mu \)  in different contexts. Great progress
has been achieved in recent years in the study of their properties with lattice QCD
methods, both in vacuum and at finite temperature [2]}. In the continuum,
however, although in principle the complex of the Schwinger-Dyson (DS)
and Bethe-Salpeter (BS) equations provides an adequate framework, tremendous
challenges still remain in describing simultaneously light- and heavy-flavored
mesons in vacuum within a single model interaction–truncation
scheme , , . On the other hand, there is pressing
need for different pieces of information on properties of such mesons both in
vacuum and at finite \(T\)  and \(\mu \)  for guiding experimental proposals at existing
and forthcoming facilities. Examples are charmed-hadron production via \(\bar{p}p\) 
annihilation processes , , , \(J/\Psi \) -nuclear bound states , ,
production in heavy-ion collisions of exotic molecules like the \(\Delta \bar{D}^*\)  [3]}.
Having this in mind, in the present contribution we extend to finite \(T\)  and \(\mu \)  a model that
gives a good description of the spectrum and leptonic decay constants of pseudoscalar mesons in
vacuum [4]} . The model is based on a confining, symmetry-preserving treatment
of a vector-vector four fermion contact interaction as a representation of the gluon's two-point
Schwinger function used in kernels of DS equations, originally tuned to study the
pion [5]}.
